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Elhoucien Elqorachi - One of the best experts on this subject based on the ideXlab platform.
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A generalization of d’Alembert’s other functional equation on semigroups
Aequationes mathematicae, 2020Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup S generated by its squares and equipped with an Involutive Automorphism $$\sigma $$ σ and a multiplicative function $$\mu :S\rightarrow \mathbb {C}$$ μ : S → C such that $$\mu (x\sigma (x))=1$$ μ ( x σ ( x ) ) = 1 for all $$x\in S$$ x ∈ S , we determine the complex-valued solutions of the following functional equation $$\begin{aligned} f(xy)-\mu (y)f(\sigma (y)x)=g(x)h(y),\quad x,y\in S. \end{aligned}$$ f ( x y ) - μ ( y ) f ( σ ( y ) x ) = g ( x ) h ( y ) , x , y ∈ S .
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A Class of Functional Equations of Type d’Alembert on Monoids
Frontiers in Functional Equations and Analytic Inequalities, 2019Co-Authors: Belaid Bouikhalene, Elhoucien ElqorachiAbstract:Recently, the solutions of the functional equation f(xy) − f(σ(y)x) = g(x)h(y) obtained, where σ is an Involutive Automorphism and f, g, h are complex-valued functions, in the setting of a group G and a monoid S. Our main goal is to determine the general complex-valued solutions of the following version of this equation, viz. f(xy) − μ(y)f(σ(y)x) = g(x)h(y) where \(\mu : G\longrightarrow \mathbb {C}\) is a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ G. As an application we find the complex-valued solutions (f, g, h) on groups of equation f(xy) + μ(y)g(σ(y)x) = h(x)h(y) on monoids.
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Variants of Wilson's functional equation on semigroups.
arXiv: General Mathematics, 2019Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup $S$ generated by its squares equipped with an Involutive Automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(y)g(x),\, x,y\in S\end{equation*}
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A generalization of d'Alembert's functional equation on semigroups
arXiv: General Mathematics, 2019Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup $S$ generated by its squares equipped with an Involutive Automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equation.
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Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations
Annales Mathematicae Silesianae, 2018Co-Authors: Elhoucien Elqorachi, Ahmed RedouaniAbstract:Abstract We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations ∫Sf(xyt)dµ(t)+∫Sf(xσ(y)t)dµ(t)= 2f(x)f(y), x,y ∈ S; ∫Sf(xσ(y)t)dµ(t)-∫Sf(xyt)dµ(t)= 2f(x)f(y), x,y ∈ S; where S is a semigroup, σ is an Involutive Automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an Involutive Automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an Involutive morphism.
Axel Hultman - One of the best experts on this subject based on the ideXlab platform.
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A word property for twisted involutions in Coxeter groups
Journal of Combinatorial Theory Series A, 2019Co-Authors: Mikael Hansson, Axel HultmanAbstract:Abstract Given an Involutive Automorphism θ of a Coxeter system ( W , S ) , let I ( θ ) ⊆ W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S _ -expressions (also known as admissible sequences, reduced I θ -expressions, or involution words) for any given w ∈ I ( θ ) . This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.
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A word property for twisted involutions in Coxeter groups
arXiv: Combinatorics, 2017Co-Authors: Mikael Hansson, Axel HultmanAbstract:Given an Involutive Automorphism $\theta$ of a Coxeter system $(W,S)$, let $\mathfrak{I}(\theta) \subseteq W$ denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced $\underline{S}$-expressions (also known as admissible sequences, reduced $I_\theta$-expressions, or involution words) for any given $w \in \mathfrak{I}(\theta)$. This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.
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A word property for twisted involutions in Coxeter groups
Electronic Notes in Discrete Mathematics, 2017Co-Authors: Mikael Hansson, Axel HultmanAbstract:Abstract Given an Involutive Automorphism θ of a Coxeter system (W, S), let I ( θ ) ⊆ W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S _ -expressions for any given w ∈ I ( θ ) . This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.
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Twisted identities in Coxeter groups
Journal of Algebraic Combinatorics, 2008Co-Authors: Axel HultmanAbstract:Given a Coxeter system ( W , S ) equipped with an Involutive Automorphism θ , the set of twisted identities is $$\iota (\theta )=\{\theta(w^{-1})w\mid w\in W\}.$$ We point out how ι ( θ ) shows up in several contexts and prove that if there is no s ∈ S such that s θ ( s ) is of odd order greater than 1, then the Bruhat order on ι ( θ ) is a graded poset with rank function ρ given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or ℤ-acyclic. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincaré series of ι ( θ ), i.e. its generating function with respect to ρ , a factorisation phenomenon is discussed.
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Twisted identities in Coxeter groups
Journal of Algebraic Combinatorics, 2007Co-Authors: Axel HultmanAbstract:Given a Coxeter system (W,S) equipped with an Involutive Automorphism ?, the set of twisted identities is $$\iota (\theta )=\{\theta(w^{-1})w\mid w\in W\}.$$ We point out how ?(?) shows up in several contexts and prove that if there is no s?S such that s ?(s) is of odd order greater than 1, then the Bruhat order on ?(?) is a graded poset with rank function ? given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or ?-acyclic. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincare series of ?(?), i.e. its generating function with respect to ?, a factorisation phenomenon is discussed.
Omar Ajebbar - One of the best experts on this subject based on the ideXlab platform.
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A generalization of d’Alembert’s other functional equation on semigroups
Aequationes mathematicae, 2020Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup S generated by its squares and equipped with an Involutive Automorphism $$\sigma $$ σ and a multiplicative function $$\mu :S\rightarrow \mathbb {C}$$ μ : S → C such that $$\mu (x\sigma (x))=1$$ μ ( x σ ( x ) ) = 1 for all $$x\in S$$ x ∈ S , we determine the complex-valued solutions of the following functional equation $$\begin{aligned} f(xy)-\mu (y)f(\sigma (y)x)=g(x)h(y),\quad x,y\in S. \end{aligned}$$ f ( x y ) - μ ( y ) f ( σ ( y ) x ) = g ( x ) h ( y ) , x , y ∈ S .
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Variants of Wilson's functional equation on semigroups.
arXiv: General Mathematics, 2019Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup $S$ generated by its squares equipped with an Involutive Automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(y)g(x),\, x,y\in S\end{equation*}
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A generalization of d'Alembert's functional equation on semigroups
arXiv: General Mathematics, 2019Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:Given a semigroup $S$ generated by its squares equipped with an Involutive Automorphism $\sigma$ and a multiplicative function $\mu:S\to\mathbb{C}$ such that $\mu(x\sigma(x))=1$ for all $x\in S$, we determine the complex-valued solutions of the following functional equation.
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The Cosine–Sine functional equation on a semigroup with an Involutive Automorphism
Aequationes mathematicae, 2017Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:We determine the complex-valued solutions of the following extension of the Cosine–Sine functional equation $$\begin{aligned} f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\quad x,y\in S, \end{aligned}$$ f ( x σ ( y ) ) = f ( x ) g ( y ) + g ( x ) f ( y ) + h ( x ) h ( y ) , x , y ∈ S , where S is a semigroup generated by its squares and $$\sigma $$ σ is an Involutive Automorphism of S . We express the solutions in terms of multiplicative and additive functions.
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the cosine sine functional equation on a semigroup with an Involutive Automorphism
Aequationes Mathematicae, 2017Co-Authors: Omar Ajebbar, Elhoucien ElqorachiAbstract:We determine the complex-valued solutions of the following extension of the Cosine–Sine functional equation $$\begin{aligned} f(x\sigma (y))=f(x)g(y)+g(x)f(y)+h(x)h(y),\quad x,y\in S, \end{aligned}$$ where S is a semigroup generated by its squares and \(\sigma \) is an Involutive Automorphism of S. We express the solutions in terms of multiplicative and additive functions.
Antonella Zanna - One of the best experts on this subject based on the ideXlab platform.
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Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms
Foundations of Computational Mathematics, 2001Co-Authors: Hans Munthe-kaas, G. R. W. Quispel, Antonella ZannaAbstract:The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to Involutive Automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the Involutive Automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group. In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion. Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G .
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Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms
Foundations of Computational Mathematics, 2001Co-Authors: Hans Munthe-kaas, G. R. W. Quispel, Antonella ZannaAbstract:The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to Involutive Automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the Involutive Automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group.
Vladimir V Sergeichuk - One of the best experts on this subject based on the ideXlab platform.
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roth s solvability criteria for the matrix equations ax xˆb c and x axˆb c over the skew field of quaternions with an Involutive Automorphism q qˆ
Linear Algebra and its Applications, 2016Co-Authors: Vyacheslav Futorny, Tetiana Klymchuk, Vladimir V SergeichukAbstract:Abstract The matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X − A X B = C over a field has a solution if and only if the matrices [ A C 0 I ] and [ I 0 0 B ] are simultaneously equivalent to [ A 0 0 I ] and [ I 0 0 B ] . We extend these criteria to the matrix equations A X − X ˆ B = C and X − A X ˆ B = C over the skew field of quaternions with a fixed Involutive Automorphism q ↦ q ˆ .
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Roth's solvability criteria for the matrix equations AX−XˆB=C and X−AXˆB=C over the skew field of quaternions with an Involutive Automorphism q↦qˆ
Linear Algebra and its Applications, 2016Co-Authors: Vyacheslav Futorny, Tetiana Klymchuk, Vladimir V SergeichukAbstract:Abstract The matrix equation A X − X B = C has a solution if and only if the matrices [ A C 0 B ] and [ A 0 0 B ] are similar. This criterion was proved over a field by W.E. Roth (1952) and over the skew field of quaternions by Huang Liping (1996). H.K. Wimmer (1988) proved that the matrix equation X − A X B = C over a field has a solution if and only if the matrices [ A C 0 I ] and [ I 0 0 B ] are simultaneously equivalent to [ A 0 0 I ] and [ I 0 0 B ] . We extend these criteria to the matrix equations A X − X ˆ B = C and X − A X ˆ B = C over the skew field of quaternions with a fixed Involutive Automorphism q ↦ q ˆ .
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Consimilarity and quaternion matrix equations $AX-\hat{X}B=C$, $X-A\hat{X}B=C$
Special Matrices, 2014Co-Authors: Tatiana Klimchuk, Vladimir V SergeichukAbstract:L.Huang [Linear Algebra Appl. 331 (2001) 21-30] gave a canonical form of a quaternion matrix $A$ with respect to consimilarity transformations $\tilde{S}^{-1}AS$ in which $S$ is a nonsingular quaternion matrix and $\tilde{h}:=a-bi+cj-dk$ for each quaternion $h=a+bi+cj+dk$. We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations $\hat{S}^{-1}AS$ in which $h\mapsto\hat{h}$ is an arbitrary Involutive Automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations $AX-\hat{X}B=C$ and $X-A\hat{X}B=C$.
